📝 题目
例 8 设 $f\left( x\right) \in R\left\lbrack {a,b}\right\rbrack ,g\left( x\right)$ 与 $f\left( x\right)$ 只有有限个点上取值不等. 求证:
$$ g\left( x\right) \in R\left\lbrack {a,b}\right\rbrack \text{ ,且 }{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = {\int }_{a}^{b}g\left( x\right) \mathrm{d}x\text{ . } $$
💡 答案与解析
证 令 $h\left( x\right) \overset{\text{ 定义 }}{ = }g\left( x\right) - f\left( x\right)$ ,则 $h\left( x\right)$ 只在有限个点上取非零值,所以
$$ h\left( x\right) \in R\left\lbrack {a,b}\right\rbrack ,\;\text{ 且 }\;{\int }_{a}^{b}h\left( x\right) \mathrm{d}x = 0. $$
又 $g\left( x\right) = f\left( x\right) + h\left( x\right)$ ,故有 $g\left( x\right) \in R\left\lbrack {a,b}\right\rbrack$ ,且
$$ {\int }_{a}^{b}g\left( x\right) \mathrm{d}x = {\int }_{a}^{b}f\left( x\right) \mathrm{d}x + {\int }_{a}^{b}h\left( x\right) \mathrm{d}x = {\int }_{a}^{b}f\left( x\right) \mathrm{d}x. $$